Harmonic interpolating sequences, ${L}^{p}$ and BMO
Annales de l'Institut Fourier, Tome 28 (1978) no. 4, pp. 215-228.

Soit $\left({z}_{\nu }\right)$ une suite de points du demi-plan supérieur ; si, pour $p$ tel que $1, et pour toute suite $\left({a}_{\nu }{\right)}_{\nu \in \mathbf{N}}$ dans ${\ell }^{p}\left(\mathbf{N}\right)$ il existe une fonction $f$, intégrale de poisson d’une fonction de ${L}^{p}\left(\mathbf{R}\right)$ qui vérifie :

 ${y}_{\nu }^{1/p}f\left({z}_{\nu }\right)={a}_{\nu },\phantom{\rule{3.33333pt}{0ex}}\nu =1,2,...\phantom{\rule{2em}{0ex}}\left(*\right)$

alors nous montrons que $\left({z}_{\nu },\phantom{\rule{3.33333pt}{0ex}}\nu \in \mathbf{N}\right)$ est une suite d’interpolation pour ${H}^{\infty }$. De même, si on fait l’hypothèse qu’il existe une solution $f$, intégrale de Poisson d’une fonction de BMO qui vérifie $\left(*\right)$ avec $p=+\infty$ et $\left({a}^{\nu }\right)$ dans ${\ell }^{\infty }\left(\mathbf{N}\right)$, $\left({z}_{\nu }\right)$ est encore une suite d’interpolation pour ${H}^{\infty }$.

Un théorème un peu plus général est prouvé et on donne un contre-exemple dans le cas où $p\le 1$.

Let $\left({z}_{\nu }\right)$ be a sequence in the upper half plane. If $1 and if

 ${y}_{\nu }^{1/p}f\left({z}_{\nu }\right)={a}_{\nu },\phantom{\rule{3.33333pt}{0ex}}\nu =1,2,...\phantom{\rule{2em}{0ex}}\left(*\right)$

has solution $f\left(z\right)$ in the class of Poisson integrals of ${L}^{p}$ functions for any sequence $\left({a}_{\nu }\right)\in {\ell }^{p}$, then we show that $\left({z}_{\nu }\right)$ is an interpolating sequence for ${H}^{\infty }$. If $f\left({z}_{\nu }\right)={a}_{\nu }$, $\nu =1,2,...$ has solution in the class of Poisson integrals of BMO functions whenever $\left({a}_{\nu }\right)\in {\ell }^{\infty }$, then $\left({z}_{\nu }\right)$ is again an interpolating sequence for ${H}^{\infty }$. A somewhat more general theorem is also proved and a counterexample for the case $p\le 1$ is described.

@article{AIF_1978__28_4_215_0,
author = {Garnett, John B.},
title = {Harmonic interpolating sequences, $L^p$ and BMO},
journal = {Annales de l'Institut Fourier},
pages = {215--228},
publisher = {Institut Fourier},
volume = {28},
number = {4},
year = {1978},
doi = {10.5802/aif.721},
zbl = {0377.46044},
mrnumber = {80g:30024},
language = {en},
url = {www.numdam.org/item/AIF_1978__28_4_215_0/}
}
Garnett, John B. Harmonic interpolating sequences, $L^p$ and BMO. Annales de l'Institut Fourier, Tome 28 (1978) no. 4, pp. 215-228. doi : 10.5802/aif.721. http://www.numdam.org/item/AIF_1978__28_4_215_0/

[1] Eric Amar, Interpolation Lp, to appear.

[2] D. Burkholder, R. Gundy and M. Silverstein, A maximal function characterization of the class Hp, Trans. A.M.S., 157 (1971), 137-157. | MR 43 #527 | Zbl 0223.30048

[3] L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math., 80 (1958), 921-930. | MR 22 #8129 | Zbl 0085.06504

[4] L. Carleson and J. Garnett, Interpolating sequences and separation properties, Jour. d'Analyse Math., 28 (1975), 273-299. | Zbl 0347.30032

[5] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-635. | MR 54 #843 | Zbl 0326.32011

[6] R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. A.M.S., 83 (1977), 569-645. | MR 56 #6264 | Zbl 0358.30023

[7] P. R. Duren, Theory of Hp Spaces, Academic Press, New York, 1970. | MR 42 #3552 | Zbl 0215.20203

[8] C. Fefferman and E. Stein, Hp spaces of several variables, Acta Math., 129 (1972), 137-193. | MR 56 #6263 | Zbl 0257.46078

[9] J. Garnett, Interpolating sequences for bounded harmonic functions, Indiana U. Math. J., 21 (1971), 187-192. | MR 44 #1814 | Zbl 0236.30042

[10] L. Hörmander, Lp estimates for (pluri-) subharmonic functions, Math. Scand., 20 (1967), 65-78. | Zbl 0156.12201

[11] E. M. Stein, Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton University Press, Princeton, 1972. | MR 57 #12890 | Zbl 0242.32005

[12] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970. | MR 44 #7280 | Zbl 0207.13501

[13] N. Varopoulos, Sur un problème d'interpolation, C.R. Acad. Sci. Paris, Ser. A, 274 (1972), 1539-1542. | MR 46 #2417 | Zbl 0236.41001